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Άλγεβρα Lie
Άλγεβρα Lie Lie Algebra thumb|300px| [[Άλγεβρα Lie ]] thumb|300px| [[Ομαδοθεωρία ---- Αλγεβρική Ομάδα Γενική Γραμμική Ομάδα Ορθογώνια Ομάδα Μοναδιακή Ομάδα ---- Μαθηματική Αναπαράσταση Μαθηματική Μήτρα ]] - Ένα Μαθηματικό Δόμημα. Ετυμολογία Η ονομασία "Άλγεβρα Lie" σχετίζεται ετυμολογικά με το όνομα "Lie". Εισαγωγή In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g over a field F together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto y , called the Lie bracket, satisfying the Jacobi identity. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds, with the property that the group operations of multiplication and inversion are smooth maps. Any Lie group gives rise to a Lie algebra. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to covering (Lie's third theorem). This correspondence between Lie groups and Lie algebras allows one to study Lie groups in terms of Lie algebras. Lie algebras were so termed by Hermann Weyl after Sophus Lie in the 1930s. In older texts, the name infinitesimal group is used. History Lie algebras were introduced to study the concept of infinitesimal transformations by Lie in the 1870s, and independently discovered by Wilhelm Killing in the 1880s. Definitions A Lie algebra is a vector space \mathfrak{g} over some field F'' allows more generally for a module over a commutative ring together with a binary operation \cdot,\cdot: \mathfrak{g}\times\mathfrak{g}\to\mathfrak{g} called the Lie bracket that satisfies the following axioms: * Bilinearity, :: x + b y, z = a z + b z, \quad a x + b y = ax + b y :for all scalars ''a, b'' in ''F and all elements x'', ''y, z'' in \mathfrak{g} . * Alternativity, :: x,x=0\ :for all ''x in \mathfrak{g} . * The Jacobi identity, :: [x,y,z] + [z,x,y] + [y,z,x] = 0 \ :for all x'', ''y, z'' in \mathfrak{g} . Using bilinearity to expand the Lie bracket x+y,x+y and using alternativity shows that x,y + y,x=0\ for all elements ''x, y'' in \mathfrak{g} , showing that bilinearity and alternativity together imply *Anticommutativity, :: x,y = -y,x,\ :for all elements ''x, y'' in \mathfrak{g} . If the field's characteristic is not 2 then anticommutativity implies alternativity. It is customary to express a Lie algebra in lower-case fraktur, like \mathfrak{g} . If a Lie algebra is associated with a Lie group, then the spelling of the Lie algebra is the same as that Lie group. For example, the Lie algebra of [[special unitary group|SU(''n)]] is written as \mathfrak{su}(n) . Generators and dimension Elements of a Lie algebra \mathfrak{g} are said to be generators of the Lie algebra if the smallest subalgebra of \mathfrak{g} containing them is \mathfrak{g} itself. The dimension of a Lie algebra is its dimension as a vector space over F. The cardinality of a minimal generating set of a Lie algebra is always less than or equal to its dimension. Υποσημειώσεις Εσωτερική Αρθρογραφία * Ομάδα * Ομαδοθεωρία * Ορθογώνια Ομάδα * Μοναδιακή Ομάδα * Ετεροτική Ομάδα * Αναπαράσταση * Μήτρα Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *math.upenn.edu *[ ] Κατηγορία:Μαθηματικά Δομήματα